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Theorem nodenselem3 31538
Description: Lemma for nodense 31544. If one surreal is older than another, then there is an ordinal at which they are not equal. (Contributed by Scott Fenton, 16-Jun-2011.)
Assertion
Ref Expression
nodenselem3 ((𝐴 No 𝐵 No ) → (( bday 𝐴) ∈ ( bday 𝐵) → ∃𝑎 ∈ On (𝐴𝑎) ≠ (𝐵𝑎)))
Distinct variable groups:   𝐴,𝑎   𝐵,𝑎

Proof of Theorem nodenselem3
StepHypRef Expression
1 bdayval 31494 . . . 4 (𝐵 No → ( bday 𝐵) = dom 𝐵)
21adantl 482 . . 3 ((𝐴 No 𝐵 No ) → ( bday 𝐵) = dom 𝐵)
32eleq2d 2689 . 2 ((𝐴 No 𝐵 No ) → (( bday 𝐴) ∈ ( bday 𝐵) ↔ ( bday 𝐴) ∈ dom 𝐵))
4 bdayelon 31535 . . . 4 ( bday 𝐴) ∈ On
5 nosgnn0 31504 . . . . . . . . 9 ¬ ∅ ∈ {1𝑜, 2𝑜}
6 norn 31497 . . . . . . . . . . . 12 (𝐵 No → ran 𝐵 ⊆ {1𝑜, 2𝑜})
76adantr 481 . . . . . . . . . . 11 ((𝐵 No ∧ ( bday 𝐴) ∈ dom 𝐵) → ran 𝐵 ⊆ {1𝑜, 2𝑜})
8 nofun 31495 . . . . . . . . . . . 12 (𝐵 No → Fun 𝐵)
9 fvelrn 6309 . . . . . . . . . . . 12 ((Fun 𝐵 ∧ ( bday 𝐴) ∈ dom 𝐵) → (𝐵‘( bday 𝐴)) ∈ ran 𝐵)
108, 9sylan 488 . . . . . . . . . . 11 ((𝐵 No ∧ ( bday 𝐴) ∈ dom 𝐵) → (𝐵‘( bday 𝐴)) ∈ ran 𝐵)
117, 10sseldd 3589 . . . . . . . . . 10 ((𝐵 No ∧ ( bday 𝐴) ∈ dom 𝐵) → (𝐵‘( bday 𝐴)) ∈ {1𝑜, 2𝑜})
12 eleq1 2692 . . . . . . . . . 10 ((𝐵‘( bday 𝐴)) = ∅ → ((𝐵‘( bday 𝐴)) ∈ {1𝑜, 2𝑜} ↔ ∅ ∈ {1𝑜, 2𝑜}))
1311, 12syl5ibcom 235 . . . . . . . . 9 ((𝐵 No ∧ ( bday 𝐴) ∈ dom 𝐵) → ((𝐵‘( bday 𝐴)) = ∅ → ∅ ∈ {1𝑜, 2𝑜}))
145, 13mtoi 190 . . . . . . . 8 ((𝐵 No ∧ ( bday 𝐴) ∈ dom 𝐵) → ¬ (𝐵‘( bday 𝐴)) = ∅)
1514neqned 2803 . . . . . . 7 ((𝐵 No ∧ ( bday 𝐴) ∈ dom 𝐵) → (𝐵‘( bday 𝐴)) ≠ ∅)
1615adantll 749 . . . . . 6 (((𝐴 No 𝐵 No ) ∧ ( bday 𝐴) ∈ dom 𝐵) → (𝐵‘( bday 𝐴)) ≠ ∅)
17 fvnobday 31537 . . . . . . 7 (𝐴 No → (𝐴‘( bday 𝐴)) = ∅)
1817ad2antrr 761 . . . . . 6 (((𝐴 No 𝐵 No ) ∧ ( bday 𝐴) ∈ dom 𝐵) → (𝐴‘( bday 𝐴)) = ∅)
1916, 18neeqtrrd 2870 . . . . 5 (((𝐴 No 𝐵 No ) ∧ ( bday 𝐴) ∈ dom 𝐵) → (𝐵‘( bday 𝐴)) ≠ (𝐴‘( bday 𝐴)))
2019necomd 2851 . . . 4 (((𝐴 No 𝐵 No ) ∧ ( bday 𝐴) ∈ dom 𝐵) → (𝐴‘( bday 𝐴)) ≠ (𝐵‘( bday 𝐴)))
21 fveq2 6150 . . . . . 6 (𝑎 = ( bday 𝐴) → (𝐴𝑎) = (𝐴‘( bday 𝐴)))
22 fveq2 6150 . . . . . 6 (𝑎 = ( bday 𝐴) → (𝐵𝑎) = (𝐵‘( bday 𝐴)))
2321, 22neeq12d 2857 . . . . 5 (𝑎 = ( bday 𝐴) → ((𝐴𝑎) ≠ (𝐵𝑎) ↔ (𝐴‘( bday 𝐴)) ≠ (𝐵‘( bday 𝐴))))
2423rspcev 3300 . . . 4 ((( bday 𝐴) ∈ On ∧ (𝐴‘( bday 𝐴)) ≠ (𝐵‘( bday 𝐴))) → ∃𝑎 ∈ On (𝐴𝑎) ≠ (𝐵𝑎))
254, 20, 24sylancr 694 . . 3 (((𝐴 No 𝐵 No ) ∧ ( bday 𝐴) ∈ dom 𝐵) → ∃𝑎 ∈ On (𝐴𝑎) ≠ (𝐵𝑎))
2625ex 450 . 2 ((𝐴 No 𝐵 No ) → (( bday 𝐴) ∈ dom 𝐵 → ∃𝑎 ∈ On (𝐴𝑎) ≠ (𝐵𝑎)))
273, 26sylbid 230 1 ((𝐴 No 𝐵 No ) → (( bday 𝐴) ∈ ( bday 𝐵) → ∃𝑎 ∈ On (𝐴𝑎) ≠ (𝐵𝑎)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1992  wne 2796  wrex 2913  wss 3560  c0 3896  {cpr 4155  dom cdm 5079  ran crn 5080  Oncon0 5685  Fun wfun 5844  cfv 5850  1𝑜c1o 7499  2𝑜c2o 7500   No csur 31486   bday cbday 31488
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-ord 5688  df-on 5689  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-1o 7506  df-2o 7507  df-no 31489  df-bday 31491
This theorem is referenced by:  nodenselem4  31539
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